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## Homework Statement

Prove that the set of all n x n matrices A such that AB = BA for a fixed n x n matrix B, is a subspace of M

_{nn}.

## Homework Equations

**u**+

**v**is in the same vector space as

**u**and

**v**.

*k*

**u**is in the same vector space as

**u**, where

*k*is any real number.

## The Attempt at a Solution

I am drawn to think of a diagonal matrix when I think of this question. And if I multiply a diagonal matrix by a scalar, it can only be a diagonal matrix or the zero matrix, either way, it AB still equals BA. Similarly, adding two diagonal matrices obtains either another diagonal matrix or the zero matrix...so, in this way, A is a subspace of M

_{nn}.

Am I correct?